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Math Help - Taylor Derivatives!!!!!

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    Taylor Derivatives!!!!!

    Hi! Please help me figure out up to 3 derivatives of: 25arctan(x)!!! I need the 1st, 2nd, and 3rd derivatives of this function!! Thank you so much!!!!
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    You are expected to know that \frac{d}{dx}\arctan(x) = \frac{1}{1+x^2}
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    Quote Originally Posted by elocin View Post
    Hi! Please help me figure out up to 3 derivatives of: 25arctan(x)!!! I need the 1st, 2nd, and 3rd derivatives of this function!! Thank you so much!!!!
    Why are you saying Taylor? Are you asking for the Taylor series?
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    Yes it is Taylor Series! And I know that arctan derivative is (1 over x^2+1) so is that the first derivative?? I don't know how to get the second and third.
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by elocin View Post
    Yes it is Taylor Series! And I know that arctan derivative is (1 over x^2+1) so is that the first derivative?? I don't know how to get the second and third.
    \arctan(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{2n+1}

    So we know that a_n=\frac{f^{(n)}(0)}{n!}

    So calculate the second and thir term, put it in there and solve.
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    Hmm I'm kind of confused what you put. The original problem is to find the Taylor polynomial Tn(x) for the function f at the number a of f(x)=25arctan(x) when a=1 and n=3. I know how to do Taylor series, but I am having trouble finding the second and third derivatives to plug in 1 for them. I know that like for T2, it would be f(a)+f1st deriv. of a *(x-a) + f2ndderiv. of a over 2 factorial (x-a)^2. I just need help finding the 1st, 2nd, and 3rd derivatives. Thanks!!
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by elocin View Post
    Hmm I'm kind of confused what you put. The original problem is to find the Taylor polynomial Tn(x) for the function f at the number a of f(x)=25arctan(x) when a=1 and n=3. I know how to do Taylor series, but I am having trouble finding the second and third derivatives to plug in 1 for them. I know that like for T2, it would be f(a)+f1st deriv. of a *(x-a) + f2ndderiv. of a over 2 factorial (x-a)^2. I just need help finding the 1st, 2nd, and 3rd derivatives. Thanks!!
    Ok, are you really just asking

    \frac{d}{dx}\bigg[\frac{1}{1+x^2}\bigg]=\frac{-2x}{(x^2+1)^2}

    and

    \frac{d^2}{dx^2}\bigg[\frac{1}{1+x^2}\bigg]=\frac{2(3x^2-1)}{(x^2+1)^3}
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    Yes! Thanks! But what about the 25? Does that just go away? And the first derivative is (1 over x^2+1) right?
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by elocin View Post
    Yes! Thanks! But what about the 25? Does that just go away? And the first derivative is (1 over x^2+1) right?
    .................................................. .............................What level are you in? You are doing doing Taylor series but you don't know that

    \frac{d}{dx}\bigg[cf(x)\bigg]=c\frac{d}{dx}\bigg[f(x)\bigg]

    and yes
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    Well, I wasn't sure! I get mixed up sometimes with integrals and derivatives so I was just making sure I didn't make a mistake. I'm taking calc 2 and my class ends Friday.
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by elocin View Post
    Well, I wasn't sure! I get mixed up sometimes with integrals and derivatives so I was just making sure I didn't make a mistake. I'm taking calc 2 and my class ends Friday.
    Well \int{cf(x)}dx=c\int{f(x)dx}

    Good luck on your exam! (if you have one)
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    haha Thank you! Well I know that you can pull out a constant for the normal derivatives that we did, but for Taylor series, do you do like (say for the 2nd derivative, the f(a) part) put 25f(a) of the 2nd deriv.? So would you evaluate the derivative you helped me with at a=1 and then whatever that equals, multiply it by 25? for all the derivatives? I know this is a dumb question but I think I'm confusing myself!
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  13. #13
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    Quote Originally Posted by elocin View Post
    haha Thank you! Well I know that you can pull out a constant for the normal derivatives that we did, but for Taylor series, do you do like (say for the 2nd derivative, the f(a) part) put 25f(a) of the 2nd deriv.? So would you evaluate the derivative you helped me with at a=1 and then whatever that equals, multiply it by 25? for all the derivatives? I know this is a dumb question but I think I'm confusing myself!

    What you should do is this, I showed you the second and third derivatives. So you should have this

    f(a)+f'(a)x+\frac{f''(a)x^2}{2}+\frac{f'''(a)x^3}{  6}
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    To get the 2nd and 3rd derivatives, you will need the quotient rule, which states that if h(x) = \frac{g(x)}{f(x)} then h'(x) = \frac{f(x)g'(x)-f'(x)g(x)}{f(x)^2}
    You could also (in my opinion this is preferable) use the chain rule and product rule. The product rule is \frac{d}{dx}(uv) = u\frac{dv}{dx}+v\frac{du}{dx}. Just stating the chain rule isn't very helpful for you, it really needs worked examples for you to understand it. You can find some of these here although if you have a good textbook it may well be easier to understand.
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by badgerigar View Post
    To get the 2nd and 3rd derivatives, you will need the quotient rule, which states that if h(x) = \frac{g(x)}{f(x)} then h'(x) = \frac{f(x)g'(x)-f'(x)g(x)}{f(x)^2}
    You could also (in my opinion this is preferable) use the chain rule and product rule. The product rule is \frac{d}{dx}(uv) = u\frac{dv}{dx}+v\frac{du}{dx}. Just stating the chain rule isn't very helpful for you, it really needs worked examples for you to understand it. You can find some of these here although if you have a good textbook it may well be easier to understand.
    Hopefully if she is finishing up with Calculus II that she knows the chain, product, and quotient rule.
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